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Encouraged hy Jacobi, Hennite shared with him not only the
discoveries in Abelian functions, but also sent him four tremen-
dous letters on the theory of numbers, the first early in 18-tr,
These letters, the first of which was composed when Hennite
was only twenty-four., break new ground (in what respect ^e
shall indicate presently) and are sufficient alone to establish
Hermite as a creative mathematician of the first rank. The
generality of the problems he attacked and the bold originality
of the methods he devised for their solution assure Hermite's
remembrance as one of the born arithmeticians of history.
The first letter opens with an apology. 'Nearly two years have
elapsed without my answering the letter full of goodwill which
you did me the honour to write to me. To-day I shall beg you to
pardon my long negligence and express to you all the joy I felt
in seeing myself given a place in the repertory of your works.
[Jacobi has published parts of Hermite's letter, with all due
acknowledgement, in some work of his own.] Having been for
long away from the work, I was greatly touched by such an
attestation of your kindness; allow me, Sir, to believe that it
will not desert me.9 Hermite then says that another research of
Jacobi's has inspired him to his present efforts.
If the reader will glance at what was said about uniform
functions of a single variable in the chapter on Gauss (a uniform
function takes only one value for each value of the variable),
the following statement of what Jacobi had proved should be
intelligible: a uniform function of only one variable with three
distinct periods is impossible. That uniform functions of one
variable exist having either one period or two periods is proved
by exhibiting the trigonometric functions and the elliptic func-
tions. This theorem of Jacobi's, Hermite declares, gave him his
own idea for the novel methods which he introduced into the
higher arithmetic. Although these methods are too technical
for description here, the spirit of one of them can be briefly
Arithmetic in the sense of Gauss deals with properties of the
rational integers 1,2,3, ... ; irrationals (like the square root of
2) are excluded. In particular Gauss investigated the integer
solutions of large classes of indeterminate equations in two or